Convergence True or False Proof

**ThreeLeftsAndHome** · April 26th 2010, 04:17 PM

If $a_k \to 0$ as $k \to \infty$ , and $\displaystyle\sum_{k=1}^{\infty} a_k$ converges, then $a_k \downarrow 0$ as $k \to \infty$ .

I think this is false and my example is $(-1)^k \frac{1}{k^2}$ . What you think?

**Krizalid** · April 26th 2010, 05:23 PM

Originally Posted by ThreeLeftsAndHome

If $a_k \to 0$ as $k \to \infty$ , and $\displaystyle\sum_{k=1}^{\infty} a_k$ converges, then $a_k \downarrow 0$ as $k \to \infty$ .

first information is irrelevant since given the convergence of the series, then $a_k\to0$ as $k\to\infty.$

oh, i think is misread the problem, wonder what does mean $a_k\downarrow0.$

**Defunkt** · April 26th 2010, 05:31 PM

Originally Posted by Krizalid

first information is irrelevant since given the convergence of the series, then $a_k\to0$ as $k\to\infty.$

oh, i think is misread the problem, wonder what does mean $a_k\downarrow0.$

I think it means that $a_k$ is monotone decreasing.

**Krizalid** · April 26th 2010, 05:38 PM

ah well, but it doesn't say if $a_n\ge0,$ so the counterexample is correct.

Thread: Convergence True or False Proof

LinkBack

Thread Tools

Display

Convergence True or False Proof

Similar Math Help Forum Discussions

prove whether the set property is true, false, or sometimes true and sometimes false?

Prove whether the set property is always true, always false, or sometimes true or fal

True or False. Prove if true show counter example if false.

True/False Proof

a few true or false questions about sequences/series and convergence

Search Tags